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A305007 Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k*(1 + x^k)).

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%I A305007 #11 Sep 08 2022 08:46:21
%S A305007 1,2,3,4,5,3,7,8,9,5,11,3,13,7,5,16,17,18,19,2,21,11,23,6,25,13,27,7,
%T A305007 29,5,31,32,11,17,35,36,37,19,39,20,41,21,43,11,15,23,47,12,49,50,17,
%U A305007 26,53,27,55,7,57,29,59,1,61,31,63,64,65,11,67,34,23,35,71,72,73,37,75
%N A305007 Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k*(1 + x^k)).
%F A305007 Denominators of coefficients in expansion of log(Sum_{k>=0} x^(k*(k+1)/2)) = log(Product_{k>=1} (1 - x^(2*k))/(1 - x^(2*k-1))).
%F A305007 Denominators of coefficients in expansion of log(theta_2(sqrt(x))/(2*x^(1/8))), where theta_2() is the Jacobi theta function.
%F A305007 a(n) = denominator of Sum_{d|n} (-1)^(n/d+1)/d.
%F A305007 a(n) = denominator of Sum_{d|n} (-1)^(d+1)*d/n.
%F A305007 a(n) = denominator of A002129(n)/n.
%F A305007 a(p^k) = p^k where p is a prime.
%e A305007 1, -1/2, 4/3, -5/4, 6/5, -2/3, 8/7, -13/8, 13/9, -3/5, 12/11, -5/3, 14/13, -4/7, 8/5, -29/16, 18/17, -13/18, 20/19, ...
%t A305007 nmax = 75; Rest[Denominator[CoefficientList[Series[Sum[x^k/(k (1 + x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
%t A305007 nmax = 75; Rest[Denominator[CoefficientList[Series[Log[Product[(1 - x^(2 k))/(1 - x^(2 k - 1)), {k, 1, nmax}]], {x, 0, nmax}], x]]]
%t A305007 nmax = 75; Rest[Denominator[CoefficientList[Series[Log[EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8))], {x, 0, nmax}], x]]]
%t A305007 Denominator[Table[Sum[(-1)^(n/d + 1) 1/d, {d, Divisors[n]}], {n, 75}]]
%t A305007 Denominator[Table[DivisorSum[n, -(-1)^# # &]/n, {n, 75}]]
%o A305007 (PARI) a(n) = denominator(sumdiv(n, d, (-1)^(d+1)*d/n)); \\ _Michel Marcus_, May 24 2018
%o A305007 (Magma) [Denominator(&+[(-1)^(d+1)*d/n: d in Divisors(n)]): n in [1..100]]; // _Vincenzo Librandi_, May 24 2018
%Y A305007 Cf. A002129, A010054, A017665, A017666, A098987, A098988, A305006 (numerators).
%K A305007 nonn,frac
%O A305007 1,2
%A A305007 _Ilya Gutkovskiy_, May 23 2018